Music is hierarchically structured, both in how it is perceived by listeners and how it is composed. Such structure can be captured elegantly using probabilistic grammatical models similar to those used to study natural language. They address the complexity of the structure using abstract categories in a recursive formalism. Most existing grammatical models of musical structure focus on one single dimension of music--such as melody, harmony, or rhythm. While these grammar models often work well on short musical excerpts, accurate analysis of longer pieces requires taking into account the constraints from multiple domains of structure. The present paper proposes abstract product grammars--a formalism which integrates multiple dimensions of musical structure into a single grammatical model--along with efficient parsing and inference algorithms for this formalism. We use this model to study the combination of hierarchically-structured harmonic syntax and hierarchically-structured rhythmic information. The latter is modeled by a novel grammar of rhythm that is capable of expressing temporal regularities in musical phrases. It integrates grouping structure and meter. The combined model of harmony and rhythm outperforms both single-dimension models in computational experiments. All models are trained and evaluated on a treebank of hand-annotated Jazz standards.